SECT. 3.] RELATIONS BETWEEN SEVERAL PLACES IN ORBIT. 101 



erally that p must be a positive quantity, whence the ambiguity in the deter 

 mination of the angle II Pby means of its tangent is decided; but without 

 that condition, the ambiguity may be decided at pleasure. In order that the 

 calculation may be as convenient as possible, it will be expedient to put the arbi 

 trary angle H either = A or = B or = i (A -(- B]. In the first case the equa 

 tions for determining P and p will be 



p sin ( A P) =: a, 



i A r&amp;gt;\ b acos(J3 A) 



p cos (A P) = -- r~^ -f -f- . 



sm (B A) 

 In the second case the equations will be altogether analogous ; but in the third 



And thus if the auxiliary angle t is introduced, the tangent of which -r, P will 

 be found by the formula 



tan ( M + } B P) = tan (45 + ) tan l(B A), 

 and afterwards p by some one of the preceding formulas, in which 



. . ,.- .... I at a sin (450+f) 6 sin (45 + Q 



$ (b + a } = sin (45 + O \/ -^ ; s&amp;gt; = r /.-&amp;gt; - /&amp;gt; 



V sm2f sin f^2 coSi\/2 



al ocos(45-fO 

 - 



cos 



ITT. If jo and P are to be determined from the equations 



every thing said in II. could be immediately applied provided, only, 90 -f- A 

 90 _|_ B were written there throughout instead of A and B : that their use may 

 be more convenient, we can, without trouble, add the developed formulas. The 

 general formulas will be 



p sin (B A) sin (H P) = b cos (H A)-\-a cos (H B} 

 p P i n (B A) cos (H P)= b sin (H A) a sin (H B} . 

 Thus 1 or ZT= A, they change into 



